Analytic Evaluation of Quality of Service for On-Demand Data Delivery

Transcription

1 Analytc Evaluaton of Qualty of Servce for On-Demand Data Delvery Hongfe Guo Haonan Tan ( guo@cs.wsc.edu) (haonan@cs.wsc.edu) Abstract Qualty of servce (QoS) measured as balkng probablty and average watng tme s of great nterest for on-demand data delvery servce provders. In ths study, we develop and valdate two analytcal models, a balkng model and a watng model, for evaluatng balkng probablty and average watng tme respectvely for meda servers wth lmted bandwdth. Based on n-depth analyss of smulaton results, we also propose an nterpolaton of the average servce tme n the watng model when measurng the Patchng protocol. Compared to smulaton, the balkng model captures the trend of balkng probablty wth changng server bandwdth; the watng model yelds reasonably accurate results when measurng the Herarchcal Multcast Stream Mergng protocol; the nterpolaton mproves the accuracy of the watng model when measurng the Patchng protocol. Fnally, usng those two models, we evaluate and compare two multcast protocols Patchng and HMSM and suggest server bandwdth C* as a reasonable tradeoff between QoS and server utlzaton.. Introducton Multcast network protocols are wdely used to save the server bandwdth for on-demand multmeda data delvery. As the market grows, the qualty of servce of multmeda servers gans more and more attenton from servce provders. In ths study, we develop and valdate two analytcal models, balkng model and watng model, for evaluatng balkng probablty and average watng tme respectvely for meda servers wth lmted bandwdth, whch can be effectvely used to help servce provders choose the approprate server bandwdth to optmze the cost-effectveness of the stream meda servce. Snce a server has only a fnte capacty, the ncomng meda requests are no longer contenton-free. To address the cost-effectveness ssue, t s necessary to have measures that reflect the qualty of the stream meda servce. Dfferent measures are used n dfferent models. For the balkng model where requests wthdraw whenever mmedate servce s not avalable, the customer balkng probablty s the key measure; for the watng model where clents are patent enough to wat for servce, the mean watng tme s our measure of nterest. Usng those two analytc models we compare and evaluate two multcast protocols: grace patchng and herarchcal mergng. Snce the core CMVA equatons derved here are essentally protocol ndependent, t s straghtforward to extend the modelng methodology to nclude other protocols such as dynamc skyscraper and send-latest receve earlest [HuSh97]. Several multcast protocols have been modeled analytcally, whch led to some quanttatve results on the average bandwdth requrement [EaVZb]. Yet to our knowledge, so far there are no analytc models that measure the QoS of multmeda servers wth lmted bandwdth. The man contrbuton of ths paper s manly threefold. Frstly, we develop two analytc models whch ease the decson makng durng server desgn phase by provdng QoS metrcs (balkng probablty and watng tme) that both users and servce provders concern about. Secondly, based on n-depth examnaton of estmated results aganst smulaton results, we propose an nterpolaton of the average servce tme n the watng model when measurng the Patchng protocol, whch further mproves model accuracy. Fnally, both qualtatve and quanttatve results are presented, based on whch a reasonable bandwdth C* as a trade-off between QoS and server utlzaton s suggested...

2 The rest of ths paper s organzed as follows. Secton 2 provdes the background nformaton regardng the two protocols. Secton 3 elaborates on how CMVA equatons are establshed for those two dfferent models. We present valdaton results and analyss n Secton 4. Then we compare and evaluate two multcast protocols n secton 5. Secton 6 concludes our study and gves a lst of some possble future work. 2. Two Protocols For On-Demand Data Delvery To be self-contaned, we nclude a bref overvew of the two multcast protocols used throughout our dscusson. 2. Patchng The scheme s well descrbed n [EaVZb]. In response to a gven clent request, the server delvers the requested fle n a sngle multcast stream. A clent that submts a new request for the same fle suffcently soon after ths stream has started begns lstenng to the multcast, bufferng the data receved. Each such clent s also provded a new uncast stream (.e., the patch stream) that delvers the data that the new request has mssed. Both the multcast stream and the patch stream delver data at the fle play rate. Ths requres that the clent have a receve bandwdth twce the fle play rate. The patch stream enables mmedate playback on the clent s sde. Once the playback reaches the buffered pont, the patch stream can termnate and both the old and the new requests can share the multcast stream. A common confguraton of the patchng scheme s that each stream transmts data at normal fle playback rate and the clent has a recevng bandwdth twce the playback rate so that the clent can lsten to two server streams at the same tme. Gven ths settng and nfnte server capacty, the optmzed average requred bandwdth for ths protocol s proved [EaVZb] to be: R 2N +, (2..) optmzed patchng = where N s the expected number of ncomng requests for fle durng the playback of that fle. In fact ths s a functon of server throughput on fle and the fle playback duraton T and we denote t as f(, T) for generalty. The algorthm tself s farly smple. However, the choce of the threshold s mportant to attan the optmal bandwdth because t s also a functon of N. If the request rate vares greatly from tme to tme, the threshold has to be adjusted accordngly to stay optmal. On the other hand, wth the threshold constrant, ths protocol does not take full advantage of an exstng multcast stream. It s less aggressve n stream mergng compared to herarchcal mergng, whch s descrbed next. 2.2 Herarchcal Multcast Stream Mergng (HMSM) There are two features that dstngush HMSM [EaVZa, EaVZb] from patchng. One s that every stream s a multcast stream so that any two streams for the same fle can merge; the other s that the server uses dynamc programmng to generate a merge plan n real-tme. The merge plan gets updated every tme a new stream s ntalzed for a new request to reflect the optmal scheme for all the requests that are currently served by the system. Ths algorthm has a much larger search space for optmalty compared wth patchng. The confguraton for HMSM s more flexble. But for the sake of protocol comparson, the same setup s used as n prevous secton and ths s denoted by HMSM(2, ). The bandwdth requrement for HMSM(2, ) can be reasonably well approxmated [EaVZb] by R.62 ln( N /.62 ), (2.2.) HMSM(2, ) = + whch s sgnfcantly lower when N s large. No wonder HMSM s a much more sophstcated algorthm. It employs greater server computng power to lower the bandwdth.. 2.

3 3. Customzed Mean Value Analyss (CMVA) Ths secton covers the analyss of two dfferent types of clent request behavor. In the balkng model a customer does not wat. Once a request cannot be satsfed mmedately, the customer request wll smply be dropped. In contrast, n the watng model each customer request has to wat untl t s served. Two watng requests can coalesce f they ask for the same meda fle, whch means they wll share a multcast stream when the servce becomes avalable. Therefore, coalescng also helps ease the server load and t happens regardless of the delvery protocol used by the server. Ths wll be dscussed n more detal shortly. 3. Customer Balkng Model In the balkng scenaro, there are a fxed number of streams n the server system. At any gven moment, a stream can and only can be n one of two possble states: watng state (.e. watng for customers) and actve state (.e. servng a customer). At the moment a customer arrves, f there s no watng stream, t smply leaves. Otherwse, a watng stream wll swtch to actve state, begnnng to serve the request. Once the stream termnates, t returns nto watng state. Center C streams Center /λ In order to model the balkng probablty, we restate the scenaro, yet from the streams perspectve. Intally, watng streams wat for customers n a queue. An ncomng customer actvates a stream n the queue, reducng the queue length by one. Once the stream termnates, t returns to the watng queue agan. We model ths scenaro by a closed queung network wth a fxed number of streams crcutng n t. The watng state of a stream s represented by a fcttous SSFR center, whch we call center. And the actve state of a stream s represented by a fcttous delay center, whch we call center. The SSFR center models the mean tme for a stream watng for customers. Smlarly, the delay center models the mean transmt duraton for each fle n the server. Eager et al. gave out smple formulas for the average server bandwdth used to delver stored meda fles on-demand, as a functon of clent arrval rate and fle length, accordng to dfferent protocols. Solvng those equatons teratvely, together wth the CMVA equatons, the utlzaton of the SSFR server can be calculated. We assume there are dfferent categores of fles on the server, and that the access pattern of each such category of fles follows a zpfan dstrbuton. In ths smple model, we assume there s only one category. We shall show how to extend ths model to nclude multple categores shortly. We use the followng notaton: Input Parameters C server capacty λ external customer arrval rate M number of fle categores Fgure. The balkng model. 3.

4 For =, 2,, M K the total number of dstnct fles n category T mean duraton of the entre fle n category q zpfan parameter n category P probablty of accessng category fles Output Parameters S average servce tme at center R mean resdence tme at center system throughput. For =, 2, #fles on the server p fracton of customer requests for fle C average b/w for fle S mean servce tme of fle streams at center S mean servce tme at center Q mean queue length at center throughput of streams servng fle PB mean ncomng costumer balkng probablty Requred mnmum bandwdth equaton: Gven a protocol, the average server bandwdth used to delver stored meda fles on-demand can be expressed as a functon of clent arrval rate and fle length [EaVZb]. Therefore, C = f ( p, T ), (3..) where f should be substtuted by (2..) and (2.2.) for patchng and HMSM respectvely. CMVA equaton: resdence tme at center Usng the mean value technque for queung network models, the mean R ( C ) = S ( + ) (3..2) C Q After a stream becomes the head of the queue n center (.e. at the moment an arrvng costumer actvates the former head), the average tme spent (.e. servce tme) before t can leave the center (.e. at the moment the next costumer arrves) equals the average customer nter-arrval tme. Therefore = λ S (3..3) Applyng Lttle s result, the average queue length at center Q ( R C) = (3..4). 4.

5 The system throughput C R + S = (3..5) Where the expected stream duraton S = K = p S (3..6) After an actvated stream leaves center, t serves fle wth the probablty p. Therefore, the throughput of streams servng fle = p (3..7) Applyng Lttle s result agan, the mean servce tme of fle streams at center S C = (3..8) The utlzaton of center U = (3..9) S Balkng probablty equaton: When center s beng used, we can nterpret t lke there s at least one stream watng n center. Therefore an ncomng customer wll be served. Otherwse, when center s not beng used,.e., when queue length equals to, an ncomng customer wll not fnd any server, thus wll leave wthout beng served. Therefore the costumer balkng probablty 3.2 Customer Watng Model P B = U (3..) As llustrated n Fgure 2, the watng model s an open system. The requests floatng n are represented by a Posson process. The stream meda server s modeled by a mult-channel servce center. In ths model, requests queue up when the server does not have enough bandwdth to serve ther needs. The nput parameters for ths model are the same as those for the balkng model. The output parameters can be summarzed as follows: W mean watng tme for a request (not coalesced) U system utlzaton S overall mean stream duraton estmate For =, 2,, #fles on the server p fracton of customer requests for fle S mean stream duraton for fle Q mean number of watng requests (not coalesced) for fle mean throughput of requests (not coalesced) for fle R mean resdence tme of a request (not coalesced) for fle C average number of actve streams for fle. 5.

6 R mean resdence tme adjusted for coalescng W mean watng tme adjusted for coalescng λ C channels Fgure 2. The watng model A dstnct feature of ths watng model s that requests can be coalesced f an ncomng request sees another request for the same fle watng n the queue. Consequently, the number of watng requests for a sngle fle cannot exceed one at any tme. In the steady state, the mean watng tme for each request should be the same regardless of whch fle t s askng for. In other words, all requests are equal when they are watng. For a request that cannot be coalesced wth any prevous requests, ts watng tme can be approxmated by K S W ( Q j + U C j= j C ), (3.2.) where S / C estmates the server nter-departure tme, whch characterzes how fast the queue s movng forward. S s an average over S s weghted by the correspondng. S = K K = = S, (3.2.2) Where S s can be obtaned n the same way as we dd for the balkng model. U s the server utlzaton and can be derved from C s as U K = = C C. (3.2.3) Gven the server utlzaton U, the probablty for a randomly chosen channel beng actve s also U. By assumng ndependence among channels, UC s the probablty that the server s full and cannot provde servce mmedately. However, generally ths s not the case for a mult-channel server and UC s just a heurstc to reduce the complexty of the model. Q s the mean number of watng requests that cannot coalesce for fle n the queue. Applyng Lttle s Law to the queue, we have Q W = (3.2.4) At any tme, the number of non-coalescng watng requests for a certan fle s ether zero or one. Therefore, the expected partal queue length for fle Q = Pr[ queue length for fle s one]. (3.2.5). 6.

7 By forced flow law plus the fact that any new requests for fle wll be coalesced f there s already a request for fle I watng. Hence the throughput should satsfy ( Q ) pλ =. (3.2.6) We can replace n (3.2.4) wth (3.2.6) and rearrange terms to get a more ntutve form Q Wλ =, (3.2.7) + Wλ where Wλ s the expected number of requests that coalesces wth a non-coalescng request. Thus (3.2.7) can be nterpreted as the fracton of coalescng requests for fle. Wth ths set of equatons together wth (3..) and (3..8), we are able to solve the system teratvely. Once the system s solved, t s easy to see that the mean resdence tme for a request that does not coalesce s R W + =. (3.2.8) S The remanng task s to fnd out what the mean watng tme and the mean resdence tme for all the ncomng requests should be. For a request that cannot coalesce wth prevous requests, t expects to see Wλ requests for the same fle arrve durng ts watng perod. All these requests coalesce nto a sngle request. If we assume the watng tme of the frst request to be determnstc wth value W, the mean watng tme for those coalescng requests would be W/2, thus the overall average watng tme could be computed as W W Wλ + W = 2 + Wλ. (3.2.9) Unfortunately, the mean watng tme for the frst request s a random varable nstead of a fxed number. Ths equaton only gves an approxmaton of the overall watng tme and we expect t to be more accurate when the varance of watng tme dstrbuton s small. Smlarly, f we assume the watng tme dstrbuton to be exponental (or rather, a dstrbuton wth a coeffcent of varaton of ), the resdual watng tme for each coalescng request s stll expected to be W. Therefore, W = W (3.2.) could be used as an alternatve to approxmate the overall watng tme. In general, the second moment of the watng tme dstrbuton s needed to quantfy W more accurately. For smplcty, n ths paper we only dscuss the above two approxmatons. Once W s calculated, the mean resdence tme for these requests can be modfed as R = W + S. (3.2.) Note that the formula s vald only when coalescng truly occurs,.e., λ W. In case the arrval rate s too low, coalescng s not lkely and (3.2.8) would be more approprate for the mean resdence tme estmate.. 7.

8 3.3 A Smple Interpolaton We consder usng protocol-specfc nformaton to fne-tune the analytcal model for Patchng. In fact, when a request wats for too long (longer than the threshold), the server has to open a new stream that lasts for T, whch may make the stream duraton devate greatly from the estmated value. In order to reflect ths n the model, a smple nterpolaton on stream duraton s proposed: S g( ) = S W + g( ) W + T W + g( ), (4..) where g() s the optmzed threshold. Intutvely, the nterpolated mean stream duraton s smply a weghted average between the duraton obtaned n the prevous secton and the full fle playback duraton. The watng tme and the optmal threshold are used as weghts. The full playback duraton would be favored more f the watng tme s longer than the threshold, and vce versa. 4. Valdaton We valdate our models aganst some exstng smulaton results [EaVZ99]. The comparson results show that the balkng model captures the trend of balkng probablty wth changng server bandwdth; the watng model yelds reasonably accurate results when measurng the Herarchcal Multcast Stream Mergng protocol; the nterpolaton mproves the accuracy of the watng model when measurng the Patchng protocol. 4. Balkng Model Fgure 3 s the result for the balkng model. Dotted lnes are smulaton outcomes and sold lnes are CMVA solutons. The server has 2 fles wth equal lengths. The total request arrval rate s 2 per fle playback duraton. Fle request frequences conform to the Zpfan() dstrbuton. Ths confguraton wll be used throughout the valdaton. Balkng Probablty Patchng.4 HMSM(2,).3.2 Sm-P. Sm-H.5..5 Avalable Server Bandwdth per Clent Fgure 3. Balkng Model Valdaton It can be seen that although the analytcal model captures the trend of the balkng probablty as server bandwdth ncreases, t consstently overestmates the balkng probablty when the bandwdth gets suffcently large. In some cases the relatve error could be huge. Ths s somewhat surprsng because ths. 8.

9 model should be exact. We cannot come up wth an explanaton at the moment ths paper s due. For the tme beng, we can consder the model to be at least qualtatvely vald Watng Model The mean watng tme s gven n terms of the percentage of the full playback duraton of a fle. Fgure 4 gves a comparson between watng tme approxmaton (3.2.9) and (3.2.). It appears that the latter s closer to the smulaton results. For HMSM, both approaches perform reasonably well. In contrast, nether way s satsfactory for grace patchng when the bandwdth s relatvely low. Due to the fact that request coalescng can only reduce the mean watng tme, we suspect that the model has overlooked some mportant aspect of the patchng protocol. (a) (b) Mean Watng Tme.3.2 Mean Watng Tme Avalable Server Bandwdth per Clent Patchng Sm-P HMSM(2,) Sm-H.5..5 Avalable Server Bandwdth per Clent Patchng Sm-P Fgure 4. Watng Model Valdaton (a) Determnstc Approxmaton (b) Exponental Approxmaton HMSM(2,) Further comparson of varous server statstcs revealed two sources of errors. One s the mean stream duraton for each fle. Table shows the dscrepancy between the analytcal model and the smulator when the server bandwdth s.5 channel/request. For concseness, only the overall measure and measures for the three most frequently vsted fles are lsted. The model seems to underestmate ths quantty by up to 74% for an ndvdual fle and by 56% overall. Table. Mean Stream Duraton Comparson at.5 channel/request Sm-H Duraton Access Probablty Model Smulator Fle Fle Fle Overall Interpolaton The effect of the smple servce tme nterpolaton s shown n Fgure 5 and the mprovement s sgnfcant. However, there s stll room for further mprovement. Another source of error comes from the probablty that the server s full, whch s approxmated by UC. Smulaton results suggest that ths heurstc sometmes can be off by 2% to 3%. Although other more sophstcated mechansms exst to model ths probablty [Kle76], we would rather not do so because we cannot further mprove our result sgnfcantly. 9.

10 even f we use the value obtaned from the smulator when the server load s hgh. The average number of actve streams s underestmated (as well as the server utlzaton) and the mean throughput s overestmated. Our speculaton s that the formula for bandwdth calculaton has to be adjusted to some extent when beng adapted to our model. There s no proof for ths though. Mean Watng Tme (% fle playback duraton).5.4 orgnal nterpolaton.3 smulaton Avaable Server Bandwdth per Clent 5. Protocol Evaluaton Fgure 5. Servce Tme Interpolaton for Optmzed Patchng We desgned and conducted a seres of experments usng those two analytc models to compare and evaluate Patchng protocol and HTSM protocol. Not surprsngly, the experment results show sgnfcant mpact of dfferent multcast protocols and other system parameters on the qualty of servce, whch can serve as a gudelne for choosng approprate server bandwdth. ( Note the data presented below are calculated usng the equatons derved n secton 3. and secton 3.2. ) Fgure 6 llustrates the outcome for the balkng model. By comparng two plots vertcally, we notce a left shft of curves when HMSM s used nstead of patchng, whch ndcates ts lower consumpton of server bandwdth. Wthn a sngle plot, multple curves show that the requrement on bandwdth grows much more slowly than the ncrease n number of meda fles on server. The damond shaped ponts are the sum of the average server bandwdths requred for mmedate delvery of each fle [EaVZ99] and we use C* to denote ths. For example, for a server wth 8 fles (other confguratons agree wth our default settng), C* s 396 for patchng and 29 for HMSM(2, ). These spots are strategc ponts n that the balkng probabltes at these ponts are already reasonably low. Addtonal bandwdths slghtly beyond (say wthn % above) these ponts can stll be very effectve n reducng customer balkng. But further ncrease of the server bandwdth would be much less frutful snce the balkng probablty curves flatten out really quckly...

11 (a) Balkng Probablty fles 4 fles 8 fles 6 fles Sum of Avg. Req d B/W Avalable Server Bandwdth per Clent (b). Balkng Probablty fles 4 fles 8 fles 6 fles Sum of Avg. Req d B/W Avalable Server Bandwdth per Clent Fgure 6. The Impact of Protocols on Bulkng Probabltes (a) Patchng (b) HMSM From a stream meda servce provder s perspectve, t s always desrable to maxmze the server utlzaton. To address ths ssue, Fgure 7 plots the utlzaton as a functon of the server bandwdth, dependng on the multcast protocol used and the number of fles on the server. Agan, we can fnd the damond-shaped spots remarkable: the server operates at these bandwdths wth hgh utlzaton and further nvestment n bandwdth can degrade utlzaton drastcally...

12 (a) Server Utlzaton fles 4 fles 8 fles 6 fles Sum of Avg. Req d B/W Avalable Server Bandwdth per Clent (b) Server Utlzaton fles 4 fles 8 fles 6 fles Sum of Avg. Req d B/W Avalable Server Bandwdth per Clent Fgure 7. The Impact of Protocols on Server Utlzaton (a) Patchng (b) HMSM Smlarly, the assocaton between mean request watng tme and server bandwdth s presented n Fgure 8. Our fndngs are consstent wth those for the balkng model. For the same reason, we omt the utlzaton analyss for the watng model.. 2.

13 (a) Mean Clent Watng Tme (% of playback duraton) fles 4 fles 8 fles Avalable Server Bandwdth per Clent 6 fles Sum of Avg. Req d B/W (b) Mean Clent Watng Tme (% of playback duraton) fles 4 fles 8 fles 6 fles Sum of Avg. Req d B/W Avalable Server Bandwdth per Clent Fgure 8. Impact of Protocols on Mean Watng Tme (a) Patchng (b) HMSM Fgure 9 mght be more relevant to the cost-effectveness argument: the hgher the load on the server, the more customers to lose, and vce versa. The tradeoff s always there. The trend does not vary much gven dfferent numbers of meda fles. The argument s also applcable to the watng model (not shown). Ths s essentally a gudelne for system desgners to choose the approprate server bandwdth accordng to ther desgn polces.. 3.

14 Server Utlzaton fles.4.3 Sum of Avg. Req d B/W Balkng Probablty 2 fles 4 fles 8 fles Fgure 9. The Tradeoff: Utlzaton vs. Balkng 6. Concluson And Future Work In ths study, we use CMVA technques to develop two analytcal models, balkng model and watng model, for evaluatng balkng probablty and average watng tme for meda servers wth lmted bandwdth. Based on n-depth analyss of smulaton results, we also propose an nterpolaton of the average servce tme n the watng model when measurng the Patchng protocol. Compared to smulaton, although the balkng model captures the trend of balkng probablty wth changng server bandwdth, t can ncur a large relatve error when the server utlzaton s low. The watng model performs reasonably well for the HMSM protocol, yet for patchng, t demonstrates some pecularty. The smple servce tme nterpolaton yelds a more accurate estmate of the mean stream duraton, whch n turn helps the estmaton of the mean watng tme. Based on evaluaton results, t s reasonable to take C* as sweet spots n the system desgn space. And as we expected, HMSM s always more effectve than patchng n bandwdth savng. Future work can be conducted from the followng aspects to further mprove and complete ths study: Investgatng why the balkng model predctons devate from smulaton results when the server capacty gets large. Further model debuggng on the watng model for the optmzed patchng protocol. Applyng the analytcal models to meda servers wth multple categores of fles (dfferent playback duratons, dfferent populartes and so forth).. 4.

15 Acknowledgements Specal thanks to Professor Mary Vernon for all the help she gave, and to Derek Eager for provdng us wth the smulator. References [CaLo97] Carter, S. and Long, D., Improvng Vdeo-on-Demand Server Effcency Through Stream Tappng, Proc. 6 th ICCCN 97, Las Vegas, NV, Sept. 997, pp [EaVZ99] Eager, D., Vernon, M., and Zahorjan, J., Optmal and Effcent Mergng Schedules for Vdeoon-Demand Servers, Porc. 7 th ACM Int l. Multmeda Conf. (ACM MULTIMEDIA 99), Orlando, FL, Nov. 999, pp [EaVZa] Eager, D., Vernon, M., and Zahorjan, J., "Bandwdth Skmmng: A Technque for Cost- Effectve Vdeo-on-Demand", Proc. Multmeda Computng and Networkng 2 (MMCN'), San Jose, CA, January 25-27, 2. [EaVZb] Eager, D., Vernon, M., and Zahorjan, J., "Mnmzng Bandwdth Requrements for On- Demand Data Delvery", Techncal Report #48a, Computer Scence Dept., Unversty of Wsconsn. [GoLM96] Golubchk, L., Lu, J., and Muntz, R., Adaptve Pggybackng: A Novel Technque for DataSharng n VdeoOn -Demand Storage Servers, ACM Multmeda Systems Journal, 4(3):4--55, 996. [HuSh97] Hua, K., and Sheu, S., Skyscraper Broadcastng: A New Broadcastng Scheme for Metropoltan Vdeo-on-Demand Systems, Proc. ACM SIGCOMM 97 Conf., Cannes, France, Sept. 997, pp [Kle76] Klenrock, L., Queueng Systems, vol., John Wley & Sons,